The answer is in the FL Statutes: Section 733.304 - A person who is not domiciled in the state cannot qualify as personal representative unless the person is: (1) A legally adopted child or adoptive parent of the decedent; (2) Related by lineal consanguinity to the decedent; (3) A spouse or a brother, sister, uncle, aunt, nephew, or niece of the decedent, or someone related by lineal consanguinity to any such person; or (4) The spouse of a person otherwise qualified under this section.
Please note, Sarasota County will require a bond be posted since you are not a FL resident. If you're looking to prepare a FL Will, contact a local attorney to ensure you've got it all lined up correctly.
A lot of states have laws like this and a FL attorney will do a better job at answering the question for you than I can, but in NC, an individual from out of state may serve as executor, so long as they have appointed someone in NC to accept service of process in any proceeding as it relates to the estate. I would like to believe that the good people of Florida have a similar exception to the rule. Good Luck.